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Subjects:

Statistical inference, t-test, Statistical Analysis, Probability and Statistical Inference, Probability and Statistics

Level:

Bachelors/Undergraduate, Masters/Postgraduate

Types:

Assessment, Exam, Handwritten Notes, Homework, Past paper discussion

Language used:

English

The pdf includes handwritten solutions to the following questions.

**CW 19.10. CPI:** Table 1 gives the consumer price index (CPI) for the UK, US, France, and Germany from 2009 to 2018. CPI is a measure of the change in time of the price paid by the customer for a basket of fixed goods and services; here it has been standardized to be 100 in 2015. The data were provided by the OECD and were obtained from https://data.oecd.org/price/inflation-cpi.htm.

Assume that the data for the US are an IID sample from a Normal(µ1, σ2) distribution and that the data for Germany are an IID sample from a Normal(µ2, σ2) distribution. We wish to test whether or not the mean CPI for the US is the same as the mean CPI for Germany.

**(a)** Explain why a paired approach is appropriate for this test.

**(b)** Calculate the differences di = xi – yi.

**(c)** Using your answer to part (b) or otherwise, perform a t-test at the 10% level to determine whether or not there is evidence that the mean CPI for the US is the same as the mean CPI for Germany. You should state your hypothesis and p-value clearly.

Year France Germany United Kingdom United States

********* 90.5

********* 92

********* 94.9

********* 6.9

********* 98.3

********* 99.9

********* 100

********* 01 101.3

********* 3.6 103.4

********* 06 105.9

Table 1: Consumer price index (CPI) from 2009 to 2018 for four Western countries. CPI has been standardized to 100 in 2015.

**CW 19.11. The fuel efficiency of Prius**: Fueleconomy.gov, the official US government source for fuel economy information, allows users to share gas mileage information on their vehicles.

Mileage (in MPG)

(The Histogram is given here.)

The histogram shows the distribution of gas mileage in miles per gallon (MPG) from 14 users who drive a 2012 Toyota Prius. The sample mean is 53.3 MPG and the standard deviation is 5.2 MPG. Note that these data are user estimates and since the source data cannot be verified, the accuracy of these estimates is not guaranteed.

**(a) **We would like to use this data to evaluate the average gas mileage of all 2012 Prius drivers. Do you think this is reasonable? Why or why not?

**(b)** The EPA claims that a 2012 Prius gets 50 MPG (city and highway mileage combined). Do these data provide strong evidence against this estimate for drivers who participate on fueleconomy.gov? Note any assumptions you must make as you proceed with the test.

**(c)** Calculate a 95% confidence interval for the average gas mileage of a 2012 Prius by drivers who participate on fueleconomy.gov.

**CW 19.12**. **Life rating in Greece**: Greece has faced a severe economic crisis since the end of 2009. A Gallup poll surveyed 1,000 randomly sampled Greeks in 2011 and found that 25% of them said they would rate their lives poorly enough to be considered “suffering”.

**(a)** Describe the population parameter of interest. What is the value of the point estimate of this parameter?

(b) Check if the conditions required for constructing a confidence interval based on these data are met.

**(c)** Construct a 95% confidence interval for the proportion of Greeks who are “suffering".

**(d)** What does “95% confidence" mean?

**(e)** Now calculate a 99% confidence interval for the same parameter and interpret it in the context of the data.

**(f)** Compare the widths of the 95% and 99% confidence intervals. Which one is wider? Explain.

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